College Algebra, Chapter 1, 1.5, Section 1.5, Problem 74

Solve the equation $\sqrt{x} + a \sqrt[3]{x} + b \sqrt[6]{x} + ab = 0$. Suppose that $a$ and $b$ are positive real number.


$\begin{equation} \begin{aligned} \sqrt{x} + a \sqrt[3]{x} + b \sqrt[6]{x} + ab =& 0 && \text{Given} \\ \\ (\sqrt{x} + a \sqrt[3]{x}) +(b \sqrt[6]{x} + ab) =& 0 && \text{Group terms} \\ \\ \sqrt[3]{x} (\sqrt[6]{x} + a) + b (\sqrt[6]{x} + a) =& 0 && \text{Factor out } \sqrt[3]{x} \text{ and } b \\ \\ (\sqrt[3]{x} + b)(\sqrt[6]{x} + a) =& 0 && \text{Factor out } \sqrt[3]{x} + b \\ \\ \sqrt[3]{x} + b =& 0 \text{ and } \sqrt[6]{x} + a = 0 && \text{Zero Product Property} \\\ \\ x =& (-b)^3 \text{ and } x = (-a)^6 && \text{Solve for } x \\ \\ x =& -b^3 \text{ and } x = a^6 && \end{aligned} \end{equation}$